Solving Laser Diffraction with Helium-Neon Laser

[G01]

A helium neon laser ($$\lambda$$ = 633nm. is built out of a glass tube 1.0mm, 1 X 10^-3m, in diameter. One mirror is partialy transparent letting the laser beam out. An electrical discharge causes the tube to glow like a neon light. From ans optical perspective, the laser beam is a light wave that diffracts out through a 1.0mm diameter circular opening.

a) can the beam ever be perfectly parallel? No because there will always be diffraction.
b) Whst is the minimum divergence angle, $$\theta_1$$, of the beam.

This is a circular aperture so the angle of divergence is:

$$\frac{1.22\lambda}{D}$$ where D is the diameter of the aperture.

So the angle is $$\theta_1 = \frac{1.22(633X10^{-9})}{1X10^{-3}} = 7.7X10^{-4} radians.$$

The above answer is apparently wrong but I can't figure out why.

c) What is the diameter of the laser beam after it has traveled 3.0m

$$w = \frac{2.44\lambda L}{D} = .004m$$
This is also wrong. The actual answer is .002m, but again. I don't know why.

d) the diamter after 1.0km? This involes the same method as part c and agin I got it wrong. What am I missing here that's screwing me up. Thank you for the help.

 

[AvroArrow]

In part c and d you are using the wrong formula. The formula for calculating the diameter of a beam after travelling a distance L isw = \frac{2.44\lambda L}{D^2}So in part c, the diameter of the laser beam after it has travelled 3.0m is w = \frac{2.44(633X10^{-9}) 3}{1X10^{-6}} = .002mand in part d, the diameter after 1.0km is w = \frac{2.44(633X10^{-9}) 1000}{1X10^{-6}} = 0.2m

 

[kyle8921]

a) As mentioned in the question, the beam will always experience diffraction, so it can never be perfectly parallel.

b) The formula for the angle of divergence is correct. However, the value of lambda given in the question is in meters, not nanometers. So the correct calculation would be:
\theta_1 = \frac{1.22(633X10^{-9})}{1X10^{-3}} = 7.7X10^{-7} radians.

c) The formula for the diameter of the laser beam after traveling a distance L is:
w = \frac{2.44\lambda L}{D}
Substituting the given values, we get:
w = \frac{2.44(633X10^{-9})3.0}{1X10^{-3}} = 7.3X10^{-4} meters.

d) Using the same formula as part c, we get:
w = \frac{2.44(633X10^{-9})1.0X10^{3}}{1X10^{-3}} = 1.53 meters.

I believe the errors in your calculations were due to incorrect units and using the wrong value for lambda. Make sure to double check your units and values when using formulas.